June  2020, 14(3): 463-487. doi: 10.3934/ipi.2020022

Uniqueness and stability for the recovery of a time-dependent source in elastodynamics

1. 

School of Mathematical Sciences, Nankai University, Tianjin, China, 300071

2. 

Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

* Corresponding author

Received  July 2019 Revised  December 2019 Published  March 2020

This paper is concerned with inverse source problems for the time-dependent Lamé system in an unbounded domain corresponding to either the exterior of a bounded cavity or the full space $ {\mathbb{R}}^3 $. If the time and spatial variables of the source term can be separated with compact support, we prove that the vector valued spatial source term can be uniquely determined by boundary Dirichlet data in the exterior of a given cavity. Uniqueness and stability for recovering some class of time-dependent source terms are also obtained by using, respectively, partial and full boundary data.

Citation: Guanghui Hu, Yavar Kian. Uniqueness and stability for the recovery of a time-dependent source in elastodynamics. Inverse Problems & Imaging, 2020, 14 (3) : 463-487. doi: 10.3934/ipi.2020022
References:
[1]

K. Aki and P. G. Richards, Quantitative Seismology, 2nd edition, University Science Books, Mill Valley: California, 2002. Google Scholar

[2]

H. AmmariE. BretinJ. Garnier and A. Wahab, Time-reversal algorithms in viscoelastic media, European J. Appl. Math., 24 (2013), 565-600.  doi: 10.1017/S0956792513000107.  Google Scholar

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D. D. AngM. IkehataD. D. Trong and M. Yamamoto, Unique continuation for a stationary isotropic Lamé system with variable coefficients, Comm. Partial Differential Equations, 23 (1998), 371-385.  doi: 10.1080/03605309808821349.  Google Scholar

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J. ApraizL. EscauriazaG. Wang and C. Zhang, Observability inequalities and measurable sets, J. Eur. Math. Soc. (JEMS), 16 (2014), 2433-2475.  doi: 10.4171/JEMS/490.  Google Scholar

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G. Bao, G. Hu, Y. Kian and T. Yin, Inverse source problems in elastodynamics, Inverse Problems, 34 (2018), 045009. doi: 10.1088/1361-6420/aaaf7e.  Google Scholar

[7]

G. BaoG. HuJ. Sun and T. Yin, Direct and inverse elastic scattering from anisotropic media, J. Math. Pures Appl., 117 (2018), 263-301.  doi: 10.1016/j.matpur.2018.01.007.  Google Scholar

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M. Bellassoued, Unicité et contrôle pour le système de Lamé, ESAIM Control Optim. Calc. Var., 6 (2001), 561-592.  doi: 10.1051/cocv:2001123.  Google Scholar

[9]

M. BellassouedO. Imanuvilov and M. Yamamoto, Inverse problem of determining the density and two Lamé coefficients by boundary data, SIAM J. Math. Anal., 40 (2008), 238-265.  doi: 10.1137/070679971.  Google Scholar

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M. Bellassoued and M. Yamamoto, Carleman estimate and inverse source problem for Biot's equations describing wave propagation in porous media, Inverse Problems, 29 (2013), 115002. doi: 10.1088/0266-5611/29/11/115002.  Google Scholar

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M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer Monographs in Mathematics, Springer, Tokyo, 2017. doi: 10.1007/978-4-431-56600-7.  Google Scholar

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M. Bellassoued and M. Yamamoto, Lipschitz stability in determining density and two Lamé coefficients, J. Math. Anal. Appl., 329 (2007), 1240-1259.  doi: 10.1016/j.jmaa.2006.06.094.  Google Scholar

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I. B. Aïcha, Stability estimate for hyperbolic inverse problem with time-dependent coefficient, Inverse Problems, 31 (2015), 125010. doi: 10.1088/0266-5611/31/12/125010.  Google Scholar

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M. V. de HoopL. Oksanen and J. Tittelfitz, Uniqueness for a seismic inverse source problem modeling a subsonic rupture, Comm. Partial Differential Equations, 41 (2016), 1895-1917.  doi: 10.1080/03605302.2016.1240183.  Google Scholar

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A. L. Buhgem, Multidimensional inverse problems of spectral analysis, (Russian) Dokl. Akad. Nauk SSSR, 284 (1985), 21-24.  Google Scholar

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M. Choulli and Y. Kian, Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term, J. Math. Pures Appl., 114 (2018), 235-261.  doi: 10.1016/j.matpur.2017.12.003.  Google Scholar

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M. Choulli and M. Yamamoto, Some stability estimates in determining sources and coefficients, J. Inverse Ill-Posed Probl., 14 (2006), 355-373.  doi: 10.1515/156939406777570996.  Google Scholar

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K. Fujishiro and Y. Kian, Determination of time dependent factors of coefficients in fractional diffusion equations, Math. Control Relat. Fields, 6 (2016), 251-269.  doi: 10.3934/mcrf.2016003.  Google Scholar

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G. Hu, Y. Kian, P. Li and Y. Zhao, Inverse moving source problems in electrodynamics, Inverse Problems, 35 (2019), 075001. doi: 10.1088/1361-6420/ab1496.  Google Scholar

[24]

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M. IkehataG. Nakamura and M. Yamamoto, Uniqueness in inverse problems for the isotropic Lamé system, J. Math. Sci. Univ. Tokyo, 5 (1998), 627-692.   Google Scholar

[26]

O. Y. Imanuvilov and M. Yamamoto, Carleman estimates for the non-stationary Lamé system and the application to an inverse problem, ESAIM Control Optim. Calc. Var., 11 (2005), 1-56.  doi: 10.1051/cocv:2004030.  Google Scholar

[27]

O. Imanuvilov and M. Yamamoto, Carleman estimates for the three-dimensional nonstationary Lamé system and application to an inverse problem, in Control Theory of Partial Differential Equations, Lect. Notes Pure Appl. Math., 242, Chapman & Hall/CRC, Boca Raton, FL, 2005, 337–374. doi: 10.1201/9781420028317.  Google Scholar

[28]

V. Isakov, Inverse Source Problems, Mathematical Surveys and Monographs, 34, American Mathematical Society, Providence, RI, 1990. doi: 10.1090/surv/034.  Google Scholar

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V. Isakov, Inverse obstacle problems, Inverse Problems, 25 (2009), 123002. doi: 10.1088/0266-5611/25/12/123002.  Google Scholar

[32]

V. Isakov, On uniqueness of obstacles and boundary conditions from restricted dynamical and scattering data, Inverse Probl. Imaging, 2 (2008), 151-165.  doi: 10.3934/ipi.2008.2.151.  Google Scholar

[33]

D. JiangY. Liu and M. Yamamoto, Inverse source problem for the hyperbolic equation with a time-dependent principal part, J. Differential Equations, 262 (2017), 653-681.  doi: 10.1016/j.jde.2016.09.036.  Google Scholar

[34]

D. Jiang, Z. Li, Y. Liu and M. Yamamoto, Weak unique continuation property and a related inverse source problem for time-fractional diffusion-advection equations, Inverse Problems, 33 (2017), 055013. doi: 10.1088/1361-6420/aa58d1.  Google Scholar

[35]

M. Kawashita, On the local-energy decay property for the elastic wave equation with Neumann boundary condition, Duke Math. J., 67 (1992), 333-351.  doi: 10.1215/S0012-7094-92-06712-3.  Google Scholar

[36]

Y. Kian, Stability in the determination of a time-dependent coefficient for wave equations from partial data, J. Math. Anal. Appl., 436 (2016), 408-428.  doi: 10.1016/j.jmaa.2015.12.018.  Google Scholar

[37]

Y. Kian, D. Sambou and E. Soccorsi, Logarithmic stability inequality in an inverse source problem for the heat equation on a waveguide, Appl. Anal., (2018). doi: 10.1080/00036811.2018.1557324.  Google Scholar

[38]

S. Shumin, Carleman estimates for second-order hyperbolic systems in anisotropic cases and applications. Part I: Carleman estimates, Appl. Anal., 94 (2015), 2261-2286.  doi: 10.1080/00036811.2014.983486.  Google Scholar

[39]

M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575-596.  doi: 10.1088/0266-5611/8/4/009.  Google Scholar

[40]

V. D. Kupradze, Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, Vol. 25, North Holland, Amsterdam, 1979. Google Scholar

[41]

J-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[42]

J-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. II, Dunod, Paris, 1968. Google Scholar

[43]

E. Malinnikova, The theorem on three spheres for harmonic differential forms, in Complex Analysis, Operators, and Related Topics, Oper. Theory Adv. Appl., 113, Birkhäuser, Basel, 2000, 213–220.  Google Scholar

[44] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.   Google Scholar
[45]

N. S. Nadirašvili, A generalization of Hadamard's three circles theorem, Vestnik Moskov. Univ. Ser. I Mat. Meh., 31 (1976), 39-42.   Google Scholar

[46]

K. Rashedi and M. Sini, Stable recovery of the time-dependent source term from one measurement for the wave equation, Inverse Problems, 31 (2015), 105011. doi: 10.1088/0266-5611/31/10/105011.  Google Scholar

[47] P. M. Shearer, Introduction to Seismology, 2nd edition, Cambridge University Press, New York, 2009.  doi: 10.1017/CBO9780511841552.  Google Scholar
[48]

P. Stefanov and G. Uhlmann, Recovery of a source term or a speed with one measurement and applications, Trans. Amer. Math. Soc., 365 (2013), 5737-5758.  doi: 10.1090/S0002-9947-2013-05703-0.  Google Scholar

[49]

M. Yamamoto, Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by control method, Inverse Problems, 11 (1995), 481-496.  doi: 10.1088/0266-5611/11/2/013.  Google Scholar

[50]

M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems, J. Math. Pure Appl., 78 (1999), 65-98.  doi: 10.1016/S0021-7824(99)80010-5.  Google Scholar

[51]

M. Yamamoto and X. Zhang, Global Uniqueness and stability for an inverse wave source problem for less regular data, J. Math. Anal. Appl., 263 (2001), 479-500.  doi: 10.1006/jmaa.2001.7621.  Google Scholar

show all references

References:
[1]

K. Aki and P. G. Richards, Quantitative Seismology, 2nd edition, University Science Books, Mill Valley: California, 2002. Google Scholar

[2]

H. AmmariE. BretinJ. Garnier and A. Wahab, Time-reversal algorithms in viscoelastic media, European J. Appl. Math., 24 (2013), 565-600.  doi: 10.1017/S0956792513000107.  Google Scholar

[3] H. AmmariE. BretinJ. GarnierH. KangH. Lee and A. Wahab, Mathematical Methods in Elasticity Imaging, of Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2015.  doi: 10.1515/9781400866625.  Google Scholar
[4]

D. D. AngM. IkehataD. D. Trong and M. Yamamoto, Unique continuation for a stationary isotropic Lamé system with variable coefficients, Comm. Partial Differential Equations, 23 (1998), 371-385.  doi: 10.1080/03605309808821349.  Google Scholar

[5]

J. ApraizL. EscauriazaG. Wang and C. Zhang, Observability inequalities and measurable sets, J. Eur. Math. Soc. (JEMS), 16 (2014), 2433-2475.  doi: 10.4171/JEMS/490.  Google Scholar

[6]

G. Bao, G. Hu, Y. Kian and T. Yin, Inverse source problems in elastodynamics, Inverse Problems, 34 (2018), 045009. doi: 10.1088/1361-6420/aaaf7e.  Google Scholar

[7]

G. BaoG. HuJ. Sun and T. Yin, Direct and inverse elastic scattering from anisotropic media, J. Math. Pures Appl., 117 (2018), 263-301.  doi: 10.1016/j.matpur.2018.01.007.  Google Scholar

[8]

M. Bellassoued, Unicité et contrôle pour le système de Lamé, ESAIM Control Optim. Calc. Var., 6 (2001), 561-592.  doi: 10.1051/cocv:2001123.  Google Scholar

[9]

M. BellassouedO. Imanuvilov and M. Yamamoto, Inverse problem of determining the density and two Lamé coefficients by boundary data, SIAM J. Math. Anal., 40 (2008), 238-265.  doi: 10.1137/070679971.  Google Scholar

[10]

M. Bellassoued and M. Yamamoto, Carleman estimate and inverse source problem for Biot's equations describing wave propagation in porous media, Inverse Problems, 29 (2013), 115002. doi: 10.1088/0266-5611/29/11/115002.  Google Scholar

[11]

M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer Monographs in Mathematics, Springer, Tokyo, 2017. doi: 10.1007/978-4-431-56600-7.  Google Scholar

[12]

M. Bellassoued and M. Yamamoto, Lipschitz stability in determining density and two Lamé coefficients, J. Math. Anal. Appl., 329 (2007), 1240-1259.  doi: 10.1016/j.jmaa.2006.06.094.  Google Scholar

[13]

I. B. Aïcha, Stability estimate for hyperbolic inverse problem with time-dependent coefficient, Inverse Problems, 31 (2015), 125010. doi: 10.1088/0266-5611/31/12/125010.  Google Scholar

[14]

M. V. de HoopL. Oksanen and J. Tittelfitz, Uniqueness for a seismic inverse source problem modeling a subsonic rupture, Comm. Partial Differential Equations, 41 (2016), 1895-1917.  doi: 10.1080/03605302.2016.1240183.  Google Scholar

[15]

A. L. Buhgem, Multidimensional inverse problems of spectral analysis, (Russian) Dokl. Akad. Nauk SSSR, 284 (1985), 21-24.  Google Scholar

[16]

M. Choulli and Y. Kian, Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term, J. Math. Pures Appl., 114 (2018), 235-261.  doi: 10.1016/j.matpur.2017.12.003.  Google Scholar

[17]

M. Choulli and M. Yamamoto, Some stability estimates in determining sources and coefficients, J. Inverse Ill-Posed Probl., 14 (2006), 355-373.  doi: 10.1515/156939406777570996.  Google Scholar

[18]

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Vol. 219, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[19]

M. Eller, V. Isakov, G. Nakamura and D. Tataru, Uniqueness and stability in the cauchy problem for Maxwell and elasticity systems, in Nonlinear Partial Differential Equations and their Applications. Collège de France Seminar, Vol. XIV (Paris, 1997/1998), Stud. Math. Appl., 31, North-Holland, Amsterdam, 2002, 329–349. doi: 10.1016/S0168-2024(02)80016-9.  Google Scholar

[20]

M. Eller and D. Toundykov, A global Holmgren theorem for multidimensional hyperbolic partial differential equations, Appl. Anal., 91 (2012), 69-90.  doi: 10.1080/00036811.2010.538685.  Google Scholar

[21]

K. Fujishiro and Y. Kian, Determination of time dependent factors of coefficients in fractional diffusion equations, Math. Control Relat. Fields, 6 (2016), 251-269.  doi: 10.3934/mcrf.2016003.  Google Scholar

[22]

G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, 164, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-68545-6.  Google Scholar

[23]

G. Hu, Y. Kian, P. Li and Y. Zhao, Inverse moving source problems in electrodynamics, Inverse Problems, 35 (2019), 075001. doi: 10.1088/1361-6420/ab1496.  Google Scholar

[24]

G. HuY. Kian and Y. Zhao, Uniqueness to some inverse source problems for the wave equation in unbounded domains, Acta Math. Appl. Sin. Engl. Ser., 36 (2020), 134-150.  doi: 10.1007/s10255-020-0917-4.  Google Scholar

[25]

M. IkehataG. Nakamura and M. Yamamoto, Uniqueness in inverse problems for the isotropic Lamé system, J. Math. Sci. Univ. Tokyo, 5 (1998), 627-692.   Google Scholar

[26]

O. Y. Imanuvilov and M. Yamamoto, Carleman estimates for the non-stationary Lamé system and the application to an inverse problem, ESAIM Control Optim. Calc. Var., 11 (2005), 1-56.  doi: 10.1051/cocv:2004030.  Google Scholar

[27]

O. Imanuvilov and M. Yamamoto, Carleman estimates for the three-dimensional nonstationary Lamé system and application to an inverse problem, in Control Theory of Partial Differential Equations, Lect. Notes Pure Appl. Math., 242, Chapman & Hall/CRC, Boca Raton, FL, 2005, 337–374. doi: 10.1201/9781420028317.  Google Scholar

[28]

V. Isakov, Inverse Source Problems, Mathematical Surveys and Monographs, 34, American Mathematical Society, Providence, RI, 1990. doi: 10.1090/surv/034.  Google Scholar

[29]

V. Isakov, Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, 127, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4899-0030-2.  Google Scholar

[30]

V. Isakov, A nonhyperbolic Cauchy problem for $\square_b\square_c$ and its applications to elasticity theory, Comm. Pure Appl. Math., 39 (1986), 747-767.  doi: 10.1002/cpa.3160390603.  Google Scholar

[31]

V. Isakov, Inverse obstacle problems, Inverse Problems, 25 (2009), 123002. doi: 10.1088/0266-5611/25/12/123002.  Google Scholar

[32]

V. Isakov, On uniqueness of obstacles and boundary conditions from restricted dynamical and scattering data, Inverse Probl. Imaging, 2 (2008), 151-165.  doi: 10.3934/ipi.2008.2.151.  Google Scholar

[33]

D. JiangY. Liu and M. Yamamoto, Inverse source problem for the hyperbolic equation with a time-dependent principal part, J. Differential Equations, 262 (2017), 653-681.  doi: 10.1016/j.jde.2016.09.036.  Google Scholar

[34]

D. Jiang, Z. Li, Y. Liu and M. Yamamoto, Weak unique continuation property and a related inverse source problem for time-fractional diffusion-advection equations, Inverse Problems, 33 (2017), 055013. doi: 10.1088/1361-6420/aa58d1.  Google Scholar

[35]

M. Kawashita, On the local-energy decay property for the elastic wave equation with Neumann boundary condition, Duke Math. J., 67 (1992), 333-351.  doi: 10.1215/S0012-7094-92-06712-3.  Google Scholar

[36]

Y. Kian, Stability in the determination of a time-dependent coefficient for wave equations from partial data, J. Math. Anal. Appl., 436 (2016), 408-428.  doi: 10.1016/j.jmaa.2015.12.018.  Google Scholar

[37]

Y. Kian, D. Sambou and E. Soccorsi, Logarithmic stability inequality in an inverse source problem for the heat equation on a waveguide, Appl. Anal., (2018). doi: 10.1080/00036811.2018.1557324.  Google Scholar

[38]

S. Shumin, Carleman estimates for second-order hyperbolic systems in anisotropic cases and applications. Part I: Carleman estimates, Appl. Anal., 94 (2015), 2261-2286.  doi: 10.1080/00036811.2014.983486.  Google Scholar

[39]

M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575-596.  doi: 10.1088/0266-5611/8/4/009.  Google Scholar

[40]

V. D. Kupradze, Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, Vol. 25, North Holland, Amsterdam, 1979. Google Scholar

[41]

J-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[42]

J-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. II, Dunod, Paris, 1968. Google Scholar

[43]

E. Malinnikova, The theorem on three spheres for harmonic differential forms, in Complex Analysis, Operators, and Related Topics, Oper. Theory Adv. Appl., 113, Birkhäuser, Basel, 2000, 213–220.  Google Scholar

[44] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.   Google Scholar
[45]

N. S. Nadirašvili, A generalization of Hadamard's three circles theorem, Vestnik Moskov. Univ. Ser. I Mat. Meh., 31 (1976), 39-42.   Google Scholar

[46]

K. Rashedi and M. Sini, Stable recovery of the time-dependent source term from one measurement for the wave equation, Inverse Problems, 31 (2015), 105011. doi: 10.1088/0266-5611/31/10/105011.  Google Scholar

[47] P. M. Shearer, Introduction to Seismology, 2nd edition, Cambridge University Press, New York, 2009.  doi: 10.1017/CBO9780511841552.  Google Scholar
[48]

P. Stefanov and G. Uhlmann, Recovery of a source term or a speed with one measurement and applications, Trans. Amer. Math. Soc., 365 (2013), 5737-5758.  doi: 10.1090/S0002-9947-2013-05703-0.  Google Scholar

[49]

M. Yamamoto, Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by control method, Inverse Problems, 11 (1995), 481-496.  doi: 10.1088/0266-5611/11/2/013.  Google Scholar

[50]

M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems, J. Math. Pure Appl., 78 (1999), 65-98.  doi: 10.1016/S0021-7824(99)80010-5.  Google Scholar

[51]

M. Yamamoto and X. Zhang, Global Uniqueness and stability for an inverse wave source problem for less regular data, J. Math. Anal. Appl., 263 (2001), 479-500.  doi: 10.1006/jmaa.2001.7621.  Google Scholar

Figure 1.  Radiation of a source in an inhomogeneous isotropic elastic medium in the exterior of a cavity. Suppose that the cavity $ D $ is known. The inverse problem is to determine the source term from the data measured on $ \partial B_R = \{x\in {\mathbb{R}}^3: |x| = R\} $
Figure 2.  Suppose that the data are collected on $ \omega $ and on $ \partial\Omega $. The inverse problem is to determine the value of $ g $ on $ \Omega $
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